5.4 – Use Medians and Altitudes

Median Line from the vertex of a triangle to the midpoint of the opposite side

In your group, each person draw a different sized triangle. One should be scalene obtuse, one scalene acute, scalene right, and one isosceles. Then construct the medians of the triangle.

A B C **always inside the triangle

Point of concurrency Property centroid 2/3 the distance from each vertex and 1/3 distance from the midpoint

Special Segment Definition Median Line from the vertex to midpoint of opposite side

Concurrency PropertyDefinition Centroid 2/3 the distance from each vertex and 1/3 distance from the midpoint

Altitude Line from the vertex of a triangle perpendicular to the opposite side

In your group, each person draw a different sized triangle. One should be scalene obtuse, one scalene acute, scalene right, and one isosceles. Then construct the altitudes of the triangle.

AB C

Point of concurrency Property orthocenter none

Special SegmentDefinition Altitude Line from vertex  to the opposite side

Concurrency PropertyDefinition orthocenter If obtuse – outside of triangle If right – at vertex of right angle If acute – inside of triangle

Perpendicular Bisector Circumcenter Angle Bisector Incenter MedianCentroid AltitudeOrthocenter P A M A C I C O

6

In  PQR, S is the centroid, PQ = RQ, UQ = 5, TR = 3, and SU = 2. Find the measure. Find TP. 3

In  PQR, S is the centroid, PQ = RQ, UQ = 5, TR = 3, and SU = 2. Find the measure. Find SV. 3 2

In  PQR, S is the centroid, PQ = RQ, UQ = 5, TR = 3, and SU = 2. Find the measure. Find RU =6 4

In  PQR, S is the centroid, PQ = RQ, UQ = 5, TR = 3, and SU = 2. Find the measure. Find ST

In  PQR, S is the centroid, PQ = RQ, UQ = 5, TR = 3, and SU = 2. Find the measure. Find VQ

In  ABC, G is the centroid, AE = 12, DC = 15. Find the measure. Find GE and AG. A C B F D E G 12 GE = 4 AG = 8

In  ABC, G is the centroid, AE = 12, DC = 15. Find the measure. Find GC and DG. A C B F D E G 12 GC = 10 DG = 5 15

Point L is the centroid for  NOM. Use the given information to find the value of x. OL = 5x – 1 and LQ = 4x – 5 5x – 1 = 2(4x – 5) 5x – 1 = 8x – 10 –1 = 3x – 10 9 = 3x 3 = x 5x – 1 4x – 5

Point L is the centroid for  NOM. Use the given information to find the value of x. LP = 2x + 4 and NP = 9x + 6 3(2x + 4) = 9x + 6 6x + 12 = 9x = 3x = 3x 2 = x 2x + 4 9x + 6

WS 3-7, 17-19, 34, 35 Constructing the Centroid and Orthocenter HW Problems #18 Angle bisector